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The adjoint may also be called the Hermitian conjugate or simply the Hermitian [1] after Charles Hermite. It is often denoted by A † in fields like physics , especially when used in conjunction with bra–ket notation in quantum mechanics .
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
Deformed Hermitian Yang–Mills equation; Hermitian adjoint; Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions; Hermitian form, a specific sesquilinear form; Hermitian function, a complex function whose complex conjugate is equal to the original function with the variable changed in sign
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)}
More precisely: if and are Hilbert spaces and : is a linear map then the transpose of and the Hermitian adjoint of , which we will denote respectively by and , are related. Denote by I : X → X ∗ {\displaystyle I:X\to X^{*}} and J : Y → Y ∗ {\displaystyle J:Y\to Y^{*}} the canonical antilinear isometries of the Hilbert spaces X ...
Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis
The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. [4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs. [5]